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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On compositions of the loop and suspension functors

Author: M. H. Eggar
Journal: Proc. Amer. Math. Soc. 123 (1995), 597-606
MSC: Primary 55P65; Secondary 55P62
MathSciNet review: 1218114
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Abstract: The problem studied is whether, from knowledge of the homotopy type of ${\Omega ^{{d_k}}}{\Sigma ^{{c_k}}} \cdots {\Omega ^{{d_2}}}{\Sigma ^{{c_2}}}{\Omega ^{{d_1}}}{\Sigma ^{{c_1}}}X = MX$ for suitable spaces X, one can recover the nonnegative integers ${c_1},{d_1}, \ldots ,{c_k},{d_k}$. The Betti numbers of X and ${c_1},{d_1}, \ldots ,{c_k},{d_k}$ do determine the ith Betti number of MX, but even for small k , i and for X a sphere (say) the answer is a complicated one, since it depends on parities and graded Witt numbers depending on graded Witt numbers. It is shown that k can be found and that ${c_i},{d_j}$ can always be determined up to finitely many possibilities and usually uniquely.

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Keywords: Loop, suspension, homotopy equivalence, rational homotopy theory
Article copyright: © Copyright 1995 American Mathematical Society